| L(s) = 1 | − 2-s − 0.673·3-s + 4-s − 0.0759·5-s + 0.673·6-s + 3.29·7-s − 8-s − 2.54·9-s + 0.0759·10-s − 2.89·11-s − 0.673·12-s − 3.29·14-s + 0.0511·15-s + 16-s + 4.16·17-s + 2.54·18-s − 19-s − 0.0759·20-s − 2.21·21-s + 2.89·22-s − 2.67·23-s + 0.673·24-s − 4.99·25-s + 3.73·27-s + 3.29·28-s + 5.32·29-s − 0.0511·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.388·3-s + 0.5·4-s − 0.0339·5-s + 0.274·6-s + 1.24·7-s − 0.353·8-s − 0.848·9-s + 0.0240·10-s − 0.872·11-s − 0.194·12-s − 0.880·14-s + 0.0132·15-s + 0.250·16-s + 1.01·17-s + 0.600·18-s − 0.229·19-s − 0.0169·20-s − 0.484·21-s + 0.616·22-s − 0.557·23-s + 0.137·24-s − 0.998·25-s + 0.718·27-s + 0.622·28-s + 0.988·29-s − 0.00934·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.673T + 3T^{2} \) |
| 5 | \( 1 + 0.0759T + 5T^{2} \) |
| 7 | \( 1 - 3.29T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 + 6.16T + 37T^{2} \) |
| 41 | \( 1 - 0.478T + 41T^{2} \) |
| 43 | \( 1 + 7.31T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 + 2.83T + 59T^{2} \) |
| 61 | \( 1 - 1.99T + 61T^{2} \) |
| 67 | \( 1 - 1.73T + 67T^{2} \) |
| 71 | \( 1 + 2.08T + 71T^{2} \) |
| 73 | \( 1 - 9.32T + 73T^{2} \) |
| 79 | \( 1 + 9.61T + 79T^{2} \) |
| 83 | \( 1 - 2.18T + 83T^{2} \) |
| 89 | \( 1 + 2.01T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.986814656931493007256356273196, −7.14333169730662379246889157613, −6.23056507898126353440770531853, −5.48916213288137843115216415483, −5.07929411833671650196713351910, −4.04205444963413171136143979603, −2.95907460779644220404621929039, −2.17054908135933545338486889567, −1.18558461783557089296475285049, 0,
1.18558461783557089296475285049, 2.17054908135933545338486889567, 2.95907460779644220404621929039, 4.04205444963413171136143979603, 5.07929411833671650196713351910, 5.48916213288137843115216415483, 6.23056507898126353440770531853, 7.14333169730662379246889157613, 7.986814656931493007256356273196
